Answer to Question #231353 in Chemical Engineering for pavani

Question

13)Show that the function f (z)=(z̅ )²/z,z≠0 ;0, z=0 satiesfies Cauchy-Rieman equations at z=0.Does f'(0) exist?

Answer 1

We can prove this using the theorem:

We let f(z) = u + iv be an analytic function.

1. If f 0 (z) is identically zero, then f(z) is a constant.

2. If either Re f(z) = u or Im f(z) = v is constant, then f(z) is constant. In particular, a non-constant analytic function cannot take only real or only pure imaginary values.

 3. If |f(z)| is constant or arg f(z) is constant, then f(z) is constant.

,If, f’(z) = 0, then:

Therefore, ∂u/ ∂x = ∂v/ ∂x = 0. By the Cauchy-Riemann equations, ∂v ,∂y = ∂u /∂y = 0

We determine that f (z) is a constant. It proves f’ (0) exists.